Year 2019, Volume 68 , Issue 2, Pages 1316 - 1334 2019-08-01

Spectral expansion of Sturm-Liouville problems with eigenvalue-dependent boundary conditions

Nihal YOKUŞ [1] , Esra KIR ARPAT [2]


In this paper, we consider the operator L generated in L₂(R₊) by the differential expression

l(y)=-y′′+q(x)y,x∈R₊:=[0,∞)

 and the boundary condition

((y′(0))/(y(0)))=α₀+α₁λ+α₂λ²,

 where q is a complex valued function and α_{i}∈C,[mbox]<LaTeX>\mbox{\:}</LaTeX>i=0,1,2α₂. We have proved that spectral expansion of L in terms of the principal functions under the condition

q∈AC(R₊),  lim_{x→∞}q(x)=0,  sup[e^{ε√x}|q′(x)|]<∞,  ε>0

taking into account the spectral singularities. We have also proved the convergence of the spectral expansion.
Eigenvalues, spectral singularities, principal functions, resolvent, spectral expansion.
  • Naimark, M.A., Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operator of the second order on a semi-axis, American Mathematical Society Translations Series 2, 16, (1960), 103--193.
  • Lyance, V.E., A differential operator with spectral singularities I, II, American Mathematical Society Transactions Series 2, 60, (1967), 185--225, 227--283.
  • Gasymov, M.G. and Maksudov, F.G., The principal part of the resolvent of non-selfadjoint opeerators in neighbourhood of spectral singularities, Func. Anal. Appl, 6, (1972), 185--192.
  • Maksudov, F.G. and Allakhverdiev, B.P., Spectral analysis of a new class of non-selfadjoint operators with continuous and point spectrum, Soviet Math. Dokl., 30, (1984), 566--569.
  • Adıvar, M. and Bairamov, E., Spectral properties of non-selfadjoint difference operators, Journal of Mathematical Analysis and Applications, 261(2), (2001), 461--478.
  • Bairamov, E., Çakar, Ö. and Yanık, C., Spectral singularities of the Klein-Gordon s-wave equation, Indian Journal of Pure and Applied Mathematics, 32(6), (2001), 851--857.
  • Bairamov, E. and Çelebi, A.O., Spectrum and spectral expansion for the non-selfadjoint discrete Dirac operators, The Quarterly Journal of Mathematics. Oxford Second Series, 50(200), (1999), 371--384.
  • Bairamov, E. and Karaman, Ö., Spectral singularities of the Klein-Gordon s-wave equations with and integral boundary conditions, Acta Mathematica Hungarica, 97(1--2), (2002), 121--131.
  • Krall, A.M., Bairamov, E. and Çakar, Ö., Spectrum and spectral singularities of a quadratic pencil of a Schrödinger operator with a general boundary condition, Journal of Differential Equations, 151(2), (1999), 252--267.
  • Krall, A.M., Bairamov, E. and Çakar, Ö., Spectral analysis of non-selfadjoint discrete Schrödinger operators with spectral singularities, Mathematische Nachrichten, 231, (2001), 89--104.
  • Marchenko, V.A., Expansion in eigenfunctions of non-selfadjoint singular second-order differential operators, American Mathematical Society Transactions Series 2, 25, (1963), 99.77--130.
  • Başcanbaz-Tunca, G, Spectral expasion of a non-selfadjoint differential operator on the whole axis, J.Math.Anal.Appl., 252(1), (2000), 278--297.
  • Kır Arpat, E., An eingenfunction expansion of the non-selfadjoint Sturm-Liouville operator with a singular potential, Journal of Mathematical Chemistry, 51(8), (2013), 2196--2213.
  • Bairamov, E. and Yokuş, N., Spectral singularities of Sturm-Liouville problems with eigenvalue-dependent boundary conditions, Abstract and Applied Analysis, 2009, Article ID 289596, (2009), 8 pages.
  • Yokuş, N., Principal functions of non-selfadjoint sturm-liouville problems with eigenvalue-dependent boundary conditions, Abstract and Applied Analysis, 2011, Article ID 358912, (2011), 12 pages.
Primary Language en
Journal Section Articles
Authors

Orcid: 0000-0002-5327-2312
Author: Nihal YOKUŞ

Orcid: 0000-0002-6322-5130
Author: Esra KIR ARPAT

Dates

Application Date : November 14, 2017
Acceptance Date : August 6, 2018
Publication Date : August 1, 2019

Bibtex @research article { cfsuasmas526270, journal = {Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics}, issn = {1303-5991}, eissn = {2618-6470}, address = {Communications Dergi Editörlüğü Ankara Universitesi Fen Fakültesi 06100 Tandoğan ANKARA}, publisher = {Ankara University}, year = {2019}, volume = {68}, pages = {1316 - 1334}, doi = {10.31801/cfsuasmas.526270}, title = {Spectral expansion of Sturm-Liouville problems with eigenvalue-dependent boundary conditions}, key = {cite}, author = {Yokuş, Nihal and Kır Arpat, Esra} }
APA Yokuş, N , Kır Arpat, E . (2019). Spectral expansion of Sturm-Liouville problems with eigenvalue-dependent boundary conditions . Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics , 68 (2) , 1316-1334 . DOI: 10.31801/cfsuasmas.526270
MLA Yokuş, N , Kır Arpat, E . "Spectral expansion of Sturm-Liouville problems with eigenvalue-dependent boundary conditions" . Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 (2019 ): 1316-1334 <https://dergipark.org.tr/en/pub/cfsuasmas/issue/42777/526270>
Chicago Yokuş, N , Kır Arpat, E . "Spectral expansion of Sturm-Liouville problems with eigenvalue-dependent boundary conditions". Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 (2019 ): 1316-1334
RIS TY - JOUR T1 - Spectral expansion of Sturm-Liouville problems with eigenvalue-dependent boundary conditions AU - Nihal Yokuş , Esra Kır Arpat Y1 - 2019 PY - 2019 N1 - doi: 10.31801/cfsuasmas.526270 DO - 10.31801/cfsuasmas.526270 T2 - Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics JF - Journal JO - JOR SP - 1316 EP - 1334 VL - 68 IS - 2 SN - 1303-5991-2618-6470 M3 - doi: 10.31801/cfsuasmas.526270 UR - https://doi.org/10.31801/cfsuasmas.526270 Y2 - 2018 ER -
EndNote %0 Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics Spectral expansion of Sturm-Liouville problems with eigenvalue-dependent boundary conditions %A Nihal Yokuş , Esra Kır Arpat %T Spectral expansion of Sturm-Liouville problems with eigenvalue-dependent boundary conditions %D 2019 %J Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics %P 1303-5991-2618-6470 %V 68 %N 2 %R doi: 10.31801/cfsuasmas.526270 %U 10.31801/cfsuasmas.526270
ISNAD Yokuş, Nihal , Kır Arpat, Esra . "Spectral expansion of Sturm-Liouville problems with eigenvalue-dependent boundary conditions". Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 / 2 (August 2019): 1316-1334 . https://doi.org/10.31801/cfsuasmas.526270
AMA Yokuş N , Kır Arpat E . Spectral expansion of Sturm-Liouville problems with eigenvalue-dependent boundary conditions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.. 2019; 68(2): 1316-1334.
Vancouver Yokuş N , Kır Arpat E . Spectral expansion of Sturm-Liouville problems with eigenvalue-dependent boundary conditions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics. 2019; 68(2): 1316-1334.
IEEE N. Yokuş and E. Kır Arpat , "Spectral expansion of Sturm-Liouville problems with eigenvalue-dependent boundary conditions", Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, no. 2, pp. 1316-1334, Aug. 2019, doi:10.31801/cfsuasmas.526270